I am not familiar with that stunt. The only stunts in YS that grant consequences are an endurance stunt that grants 1 mild physical, and a conviction stunt that grants 2 mild mental to resist torture. You are correct, I do not have a book in front of me, so I may be wrong, but I am fairly sure I am not.
I am well aware a severe cancels 6 stress and a mild cancels 2, the point I was making was that if your moderate was full and you needed to cancel a 4 stress hit (lets say your 2nd, 3rd, and 4th stress boxes are filled along with your moderate consequence) you would have to spend your severe. If you had 3 mild's you would only spend 2 of them (not all 6 stress worth). Also, as I mentioned each mild will give an aspect that the opponent can tag, so if you did use all 3 like a sever and suffered another attack the opponent could easily get a free +6 (instead of the +2 they would get if you used only your severe).
I am fairly certain that all the mild consequences will heal simultaneously (I do not think this is mentioned in the rules, but it is the common sense interpretation). Your body can heal from a small arm cut and a bruise at the same time, so there shouldn't be a reason why they do not heal simultaneously. Also note that most characters require some justification to start the healing process.
So, if I remember correctly you would need to spend both a stunt and an apex skill to get the benefit you need. So it would be more costly than 1 refresh. Note that Inhuman Recovery already effectively give a free extra mild and allows moderates to heal at the speed of mild's (and mild's to heal like stress) for only 2 refresh, so this doesn't seem all that good by comparison.